Problem 45 - Algebra 2

Question

Write an equation for the ellipse that satisfies each set of conditions. major axis 8 units long and parallel to \(y\) -axis, minor axis 6 units long, center at \((-3,1)\)

Step-by-Step Solution

Answer

The equation is \( \frac{(x + 3)^2}{9} + \frac{(y - 1)^2}{16} = 1 \).

Step 1: Identify the Ellipse Form

The equation of an ellipse centered at \((h, k)\) with a vertical major axis has the form \( \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. Here, the major axis is given parallel to the y-axis.

Step 2: Calculate Semi-major and Semi-minor Axes

Since the major axis is 8 units long, the semi-major axis \(a\) is half of that, so \(a = \frac{8}{2} = 4\). The minor axis is 6 units long, making the semi-minor axis \(b = \frac{6}{2} = 3\).

Step 3: Substitute into the Ellipse Equation

Substitute \(h = -3\), \(k = 1\), \(a = 4\), and \(b = 3\) into the ellipse equation form. This gives: \( \frac{(x + 3)^2}{3^2} + \frac{(y - 1)^2}{4^2} = 1 \).

Step 4: Simplify the Equation

Simplify by squaring the values of \(a\) and \(b\). Thus, the equation becomes: \( \frac{(x + 3)^2}{9} + \frac{(y - 1)^2}{16} = 1 \).

Key Concepts

Semi-Major Axis

An ellipse is defined by two main axes: the major axis and the minor axis. The semi-major axis refers to half the length of the major axis. It is the longest radius of the ellipse and extends from the center to the perimeter along the major axis. In our exercise, the major axis is given as 8 units long. Therefore, the semi-major axis is calculated by dividing the total length by 2, which results in 4 units. The semi-major axis is crucial in determining the shape of the ellipse, as it defines its vertical or horizontal stretch.

The length of the semi-major axis ( a ) directly influences the orientation of the ellipse. If it is larger than the semi-minor axis ( b ), the ellipse will be elongated in the direction of the semi-major axis. When dealing with the equation of the ellipse, the value of a is used in the denominator of the term associated with the variable that corresponds to the major axis. This relationship helps in defining the confines of the ellipse perfectly.

Semi-Minor Axis

The semi-minor axis is one of the defining elements of an ellipse and is half the length of the minor axis. This is the shorter of the two radii and stretches from the center to the edge of the ellipse along the minor axis. For the problem at hand, the minor axis measures 6 units. Therefore, halving this gives us 3 units for the semi-minor axis. Understanding this axis helps in comprehensively characterizing the geometry of the ellipse.

In the equation of an ellipse, the semi-minor axis ( b ) appears in the term associated with the variable running along the minor axis. Its role is pivotal in the shape calculations, contributing to how narrow or stretched the ellipse appears vertically or horizontally, depending on the orientation. The smaller the semi-minor axis compared to the semi-major axis, the more elongated the ellipse.

Center of Ellipse

The center of an ellipse is essentially its balance point and forms the basis for the ellipse equation format. In any standard ellipse equation, the center is represented as ( h, k ). This point dictates the positioning of the ellipse in the coordinate plane and is crucial in forming the correct ellipse equation.

In the standard form of the ellipse equation, the terms (x - h) and (y - k) highlight how the equation is shifted concerning the origin. Our exercise specifies the center at the coordinates ( -3, 1 ). Plugging these values into h and k respectively, alters the standard equation coordinates (x, y) into (x + 3, y - 1) , effectively relocating the origin point of the ellipse. This adjustment in the equation ensures that it accurately mirrors the ellipse's position and orientation on the graph.